Properties

Label 2-690-5.3-c2-0-36
Degree $2$
Conductor $690$
Sign $-0.890 + 0.454i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (−4.85 + 1.18i)5-s − 2.44·6-s + (8.99 − 8.99i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−3.67 + 6.04i)10-s + 15.1·11-s + (−2.44 + 2.44i)12-s + (−3.57 − 3.57i)13-s − 17.9i·14-s + (7.40 + 4.49i)15-s − 4·16-s + (7.84 − 7.84i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.971 + 0.237i)5-s − 0.408·6-s + (1.28 − 1.28i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.367 + 0.604i)10-s + 1.37·11-s + (−0.204 + 0.204i)12-s + (−0.274 − 0.274i)13-s − 1.28i·14-s + (0.493 + 0.299i)15-s − 0.250·16-s + (0.461 − 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.890 + 0.454i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.808357706\)
\(L(\frac12)\) \(\approx\) \(1.808357706\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (4.85 - 1.18i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (-8.99 + 8.99i)T - 49iT^{2} \)
11 \( 1 - 15.1T + 121T^{2} \)
13 \( 1 + (3.57 + 3.57i)T + 169iT^{2} \)
17 \( 1 + (-7.84 + 7.84i)T - 289iT^{2} \)
19 \( 1 + 2.57iT - 361T^{2} \)
29 \( 1 + 7.64iT - 841T^{2} \)
31 \( 1 + 55.4T + 961T^{2} \)
37 \( 1 + (-24.3 + 24.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 8.82T + 1.68e3T^{2} \)
43 \( 1 + (6.92 + 6.92i)T + 1.84e3iT^{2} \)
47 \( 1 + (45.3 - 45.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (39.7 + 39.7i)T + 2.80e3iT^{2} \)
59 \( 1 - 80.7iT - 3.48e3T^{2} \)
61 \( 1 + 74.8T + 3.72e3T^{2} \)
67 \( 1 + (-92.9 + 92.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 0.195T + 5.04e3T^{2} \)
73 \( 1 + (49.4 + 49.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 41.8iT - 6.24e3T^{2} \)
83 \( 1 + (-26.4 - 26.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 16.8iT - 7.92e3T^{2} \)
97 \( 1 + (10.2 - 10.2i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.28800381218112140526357391111, −9.080818373794358078009658955774, −7.81247514585890212168308516003, −7.36490131363606800945791505038, −6.39663117662792394267672360032, −5.03132959504273548342762694617, −4.27781999108450664013237855933, −3.46003044045894527795236662085, −1.68511221997596993769194543107, −0.61884123052847994874947290282, 1.65229443011351537405410642635, 3.43103558879560459874973732918, 4.37613449341827930615088999785, 5.12998075310030055375788677322, 5.98135434762544611451098143879, 7.07679053199193479306692621710, 8.096193377963573447524002441902, 8.723238667084915499810686432349, 9.530350884147829430806876201477, 11.10269508583081632646797605662

Graph of the $Z$-function along the critical line